
theorem Th9:
  for C be Simple_closed_curve for n be Nat st n
  is_sufficiently_large_for C holds C meets LeftComp Span(C,n)
proof
  let C be Simple_closed_curve;
  let n be Nat;
  assume
A1: n is_sufficiently_large_for C;
  then
A2: Span(C,n) is_sequence_on Gauge(C,n) by JORDAN13:def 1;
A3: 1+1 <= len Span(C,n) by GOBOARD7:34,XXREAL_0:2;
  then left_cell(Span(C,n),1,Gauge(C,n)) meets C by A1,Th7;
  then left_cell(Span(C,n),1,Gauge(C,n))\L~Span(C,n) meets C by A1,Th8,
XBOOLE_1:84;
  hence thesis by A3,A2,JORDAN9:27,XBOOLE_1:63;
end;
